PUBLICATIONS
Preprints
- Wei Zhang, Jie Fu and Chengchun Hao
On the finite-time splash singularity for the free-boundary problem of 3d incompressible neo-Hookean elastodynamics NEWIn preparation , 2025. bib@article{FHZ25,
author ={Zhang, Wei and Fu, Jie and Hao, Chengchun},
journal={},
title ={On the finite-time splash singularity for the free-boundary problem of 3d incompressible neo-Hookean elastodynamics},
year ={2025},
volume ={},
number ={},
pages ={},
} - Chengchun Hao, Tao Luo and Siqi Yang
Classification of finite-time blow-up of strong solutions to the incompressible free boundary Euler equations with surface tension NEWWe establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary. Uniquely, we make \textit{no} assumptions on symmetry, periodicity, graph representation, or domain topology (simple connectivity). At the maximal existence time $T<\infty$, up to which the velocity field and the free boundary can be continued in $H^3\times H^4$, blow-up must occur in at least one of five mutually exclusive ways:
(i) self-intersection of the free boundary for the first time;
(ii) loss of mean curvature regularity in $H^{\frac{3}{2}}$, or the free boundary regularity in $H^{2+\varepsilon}$ (for any sufficiently small constant $\varepsilon>0$);
(iii) loss of $H^{\frac{5}{2}}$ regularity for the normal boundary velocity;
(iv) the $L^1_tL^\infty$-blow-up of the tangential velocity gradient on the boundary; or
(v) the $L^1_tL^\infty$-blow-up of the full velocity gradient in the interior.
Furthermore, for simply connected domains, blow-up scenario (v) simplifies to a vorticity-based Beale-Kato-Majda criterion, and in particular, irrotational flows admit blow-up only at the free boundary.@article{HLY25,
author ={Hao, Chengchun and Luo, Tao and Yang, Siqi},
journal={},
title ={Classification of finite-time blow-up of strong solutions to the incompressible free boundary Euler equations with surface tension},
year ={2025},
volume ={},
number ={},
pages ={},
} - Wei Zhang, Jie Fu, Chengchun Hao and Siqi Yang
Global well-posedness of the free boundary problem for incompressible viscous resistive MHD in critical Besov spacesThis paper is devoted to proving the global well-posedness of the free boundary problem for the incompressible viscous resistive MHD equations. Under the Lagrangian coordinates, there exists a unique global solution in the half-space if the norm of the initial data in critical homogeneous Besov space $\dot{B}_{p, 1}^{-1+N / p}(\mathbb{R}_{+}^N)$ is small enough for $ p \in[N,2N-1)$. Inspired by the results in (J. Funct. Anal. 256: 881--927, 2009) and (J. Differ. Equations 274: 613--651, 2021) in half-space, we establish the maximal $ L^{1}$-regularity for both the Stokes equations without surface stress and the linearized equations of the magnetic field with zero boundary condition. The existence and uniqueness of the solutions of nonlinear problems is proved by using the Banach contraction mapping principle.@article{ZFHY24b,
author ={Zhang, Wei and Fu, Jie and Hao, Chengchun and Yang, Siqi},
journal={},
title ={Global well-posedness of the free boundary problem for incompressible viscous resistive MHD in critical Besov spaces},
year ={2024},
volume ={},
number ={},
pages ={},
}
- 郝成春， 现代实分析讲义， 完善中，2024.
- 郝成春、 杨思奇， 实分析讲义， 完善中，2023.
-
Chengchun Hao, Introduction to Harmonic Analysis (调和分析导论讲义, 英文版),
拟出版，2024. 目录
This book originated from lecture notes on graduate-level harmonic analysis that I have taught over the years at the Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences, as well as at Capital Normal University.
It is intended for graduate students and advanced undergraduates with a solid background in real analysis, functional analysis, and complex analysis. The book is designed to be accessible to readers at different levels, providing both depth and clarity. The background material and key notations are introduced in Appendix A, making the text approachable even for those less familiar with certain topics. Each chapter is structured with carefully chosen exercises, which are integral for reinforcing understanding and facilitating self-study. This makes the book suitable for independent learners who wish to progress at their own pace. Whether used as a textbook in formal courses or as a resource for self-guided study, the book aims to meet the needs of a wide range of learners.
The material presented draws on a variety of sources, including foundational texts as well as online lecture notes, with appropriate modifications throughout. While we do not provide extensive heuristics or motivational background, our primary goal is to present each proof in a clear, step-by-step manner to facilitate comprehension. -
Baoxiang Wang, Zhaohui Huo, Chengchun Hao and Zihua Guo
Harmonic Analysis Method for Nonlinear Evolution Equations,I, 已出版, 2011 World Scientific Pub Co Inc 300pp.
——中文版: 非线性发展方程的调和分析方法, (I)适定性理论 未出版, 2010 PDF
This monograph provides a comprehensive overview on a class of nonlinear evolution equations, such as nonlinear Schrödinger equations, nonlinear Klein–Gordon equations, KdV equations as well as Navier–Stokes equations and Boltzmann equations. The global wellposedness to the Cauchy problem for those equations is systematically studied by using the harmonic analysis methods.
This book is self-contained and may also be used as an advanced textbook by graduate students in analysis and PDE subjects and even ambitious undergraduate students. - Chengchun Hao and Siqi Yang
A priori estimates and a blow-up criterion for the incompressible ideal MHD equations with surface tension and a closed free surface NEWWe establish the a priori estimates and prove a blow-up criterion for the three-dimensional free boundary incompressible ideal magnetohydrodynamics equations. The fluid occupies a bounded region with a free boundary that is a closed surface, without assumptions of simple connectedness or periodicity of the region (thus, Fourier transforms cannot be applied), nor the graph assumption for the free boundary. The fluid is under the influence of surface tension, and flattening the boundaries using local coordinates is insufficient to resolve this problem. This is because local coordinates fail to preserve curvature, as the mean curvature of a flat boundary degenerates to zero. To address these challenges and circumvent the intricate issue of spatial regularity in Lagrangian coordinates, we utilize reference surfaces to represent the free boundary and develop new energy functionals that both preserve the material derivative and incorporate spatial-temporal scaling $\partial_t \sim \nabla^{3/2}$ in Eulerian coordinates. This method enables us to establish both low-order and high-order regularity estimates without any loss of regularity. More importantly, we prove a blow-up criterion and provide a complete classification of blow-ups, including the self-intersection of the free boundary (which the graph assumption cannot handle), the breakdown of the mean curvature, and the blow-up of the normal velocity (which Lagrangian coordinates fail to capture). To the best of our knowledge, this is the first result addressing the a priori estimates and the blow-up criterion for free boundary problems with surface tension in general regions.@article{HY25non,
author ={Hao, Chengchun and Yang, Siqi},
journal={Nonlinearity},
title ={A priori estimates and a blow-up criterion for the incompressible ideal MHD equations with surface tension and a closed free surface},
year ={2025},
volume ={38},
number ={7},
pages ={075009 (47pp)},
doi={10.1088/1361-6544/addd5a},
} - Chengchun Hao and Siqi Yang
On a regularity criterion for free boundary incompressible ideal MHD with surface tension NEWWe present the main ideas behind the proofs of a regularity criterion for the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations with surface tension in a bounded domain, as given by the authors in arXiv:2312.09473. Furthermore, we provide an overview of the latest advancements in this field.@article{HY25cis,
author ={Hao, Chengchun and Yang, Siqi},
journal={Communications in Information and Systems},
title ={On a regularity criterion for free boundary incompressible ideal MHD with surface tension},
year ={2025},
volume ={25},
number ={},
pages ={18pp},
} - Chengchun Hao and Siqi Yang
Splash singularity for the free boundary incompressible viscous MHDIn this paper, we prove the existence of smooth initial data for the two-dimensional free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity (self-intersects in at least one point) in finite time. The existence of the splash singularities is guaranteed by a local existence theorem, in which we need suitable spaces for the modified magnetic field together with the modification of the velocity and pressure such that the modified initial velocity is zero, and a stability result which allows us to construct a class of initial velocities and domains for an arbitrary initial magnetic field. It turns out that the presence of the magnetic field does not prevent the viscous fluid from forming splash singularities for certain smooth initial data.@article{HY24spvMHD,
author ={Hao, Chengchun and Yang, Siqi},
journal={Journal of Differential Equations},
title ={Splash singularity for the free boundary incompressible viscous MHD},
year ={2024},
volume ={379},
number ={},
pages ={26–103},
doi={10.1016/j.jde.2023.10.001},
} - Jie Fu, Chengchun Hao, Siqi Yang and Wei Zhang
A Beale-Kato-Majda criterion for free boundary incompressible ideal magnetohydrodynamicsWe prove a continuation criterion for the free boundary problem of three-dimensional incompressible ideal magnetohydrodynamic (MHD) equations in a bounded domain, which is analogous to the theorem given in [Beale, Kato and Majda 1984 Comm. Math. Phys. 94 61--66]. We combine the energy estimates of our previous works [Hao and Luo 2014 Arch. Ration. Mech. Anal. 212 805--847] on incompressible ideal MHD and some analogous estimates in [Ginsberg 2021 SIAM J. Math. Anal.53(3) 3366--3384] to show that the solution can be continued as long as the curls of the magnetic field and velocity, the second fundamental form and injectivity radius of the free boundary and some norms of the pressure remain bounded, provided that the Taylor-type sign condition holds.@article{FHYZ23a,
author ={Fu, Jie and Hao, Chengchun and Yang, Siqi and Zhang, Wei},
journal={Journal of Mathematical Physics},
title ={A Beale-Kato-Majda criterion for free boundary incompressible ideal magnetohydrodynamics},
year ={2023},
volume ={64},
number ={9},
pages ={091505 (16pp)},
doi={10.1063/5.0167954},
} - Chengchun Hao and Wei Zhang
A priori estimates for free boundary problem of 3D incompressible inviscid rotating Boussinesq equationsIn this paper, we consider the three-dimensional rotating Boussinesq equations (the “primitive” equations of geophysical fluid flows). Inspired by Christodoulou and Lindblad (Commu. Pure Appl. Math. 53:1536–1602, 2000), we establish a priori estimates of Sobolev norms for free boundary problem of inviscid rotating Boussinesq equations under the Taylor-type sign condition on the initial free boundary. Using the same method, we can also obtain a priori estimates for the incompressible inviscid rotating MHD system with damping.@article{HZ23b,
author ={Hao, Chengchun and Zhang, Wei},
journal={Zeitschrift für angewandte Mathematik und Physik},
title ={A priori estimates for free boundary problem of 3D incompressible inviscid rotating Boussinesq equations},
year ={2023},
volume ={74},
number ={2},
pages ={80 (21pp)},
doi={10.1007/s00033-023-01974-2},
} - Chengchun Hao and Tao Luo
Some results on free boundary problems of incompressible ideal magnetohydrodynamics equationsWe survey some recent results related to free boundary problems of incompressible ideal magnetohydrodynamics equations, and present the main ideas in the proofs of the ill-posedness in 2D when the Taylor sign condition is violated given [1], and the well-posedness of a linearized problem given in [2] in general n-dimensions $(n\geqslant 2)$ when the Taylor sign condition is satisfied and the free boundaries are closed.@article{HL22ERA,
author ={Hao, Chengchun and Luo, Tao},
journal={Electronic Research Archive},
title ={Some results on free boundary problems of incompressible ideal magnetohydrodynamics equations},
year ={2022},
volume ={30},
number ={2},
pages ={404-424},
doi={10.3934/era.2022021},
} - Chengchun Hao and Tao Luo
Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equationsThe well-posedness theory is studied for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in Lagrangian coordinates, and substitute it into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, the velocity equation is linearized with respect to the position vector field whose time derivative is the velocity. In this formulation, a key idea is to use the Lie derivative of the magnetic field taking the advantage that the magnetic field is tangential to the free boundary and divergence free. This paper contributes to the program of developing geometric approaches to study the well-posedness of free boundary problems of ideal magnetohydrodynamics equations under the condition of Taylor sign type for general free boundaries not restricted to graphs.@article{HL21JDE,
author ={Hao, Chengchun and Luo, Tao},
journal={Journal of Differential Equations},
title ={Well-posedness for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations},
year ={2021},
volume ={299},
number ={},
pages ={542-601},
doi={10.1016/j.jde.2021.07.030},
} - Chengchun Hao and Tao Luo
Ill-posedness of free boundary problem of the incompressible ideal MHDIn the present paper, we show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability $\mu _0$, in Sobolev spaces. The analysis is uniform for any $\mu_0>0$.@article{HL20CMP,
author ={Hao, Chengchun and Luo, Tao},
journal={Communications in Mathematical Physics},
title ={Ill-posedness of free boundary problem of the incompressible ideal MHD},
year ={2020},
volume ={376},
number ={1},
pages ={259–286},
doi={10.1007/s00220-019-03614-1},
} - Chengchun Hao
On the motion of free interface in ideal incompressible MHDFor the free boundary problem of the plasma–vacuum interface to 3D ideal incompressible magnetohydrodynamics, the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div–curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangential to the interface, but we do not need any restrictions on the size of the magnetic fields on the free interface. We introduce the “fictitious particle” endowed with a fictitious velocity field in vacuum to reformulate the problem to a fixed boundary problem under the Lagrangian coordinates. The $ L^2$-norms of any order covariant derivatives of the magnetic fields both in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall. The estimates of the curl of the electric fields in vacuum are also obtained, which are also indispensable in elliptic estimates.@article{Hao17ARMA,
author ={Hao, Chengchun},
journal={Archive for Rational Mechanics and Analysis},
title ={On the motion of free interface in ideal incompressible MHD},
year ={2017},
volume ={224},
number ={2},
pages ={515–553},
doi={10.1007/s00205-017-1082-7},
} - Chengchun Hao and Tao Luo
A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flowsIn the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions $ n = 2$ and $3$ by adopting a geometrical point of view used in Christodoulou and Lindblad (Commun Pure Appl Math 53:1536–1602, 2000), and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.@article{HL14ARMA,
author ={Hao, Chengchun and Luo, Tao},
journal={Archive for Rational Mechanics and Analysis},
title ={A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows},
year ={2014},
volume ={212},
number ={3},
pages ={805-847},
doi={10.1007/s00205-013-0718-5},
} - Chengchun Hao, Tao Luo and Siqi Yang
Exponential stability of a free boundary problem with spherical symmetry for a gas bubble immersed in a bounded incompressible liquid NEWCalculus of Variations and Partial Differential Equations, 64: 207 (45pp), 2025. PDFarXiv:2408.15587DOI bibThis paper is mainly concerned with the free boundary problem for an approximate model (for example, arising from the study of sonoluminescence) of a gas bubble of finite mass enclosed within a bounded incompressible viscous liquid, accounting for surface tensions at both the gas-liquid interface and the external free surface of the entire gas-liquid region. It is found that any regular spherically symmetric steady-state solution is characterized by a positive root of a ninth-degree polynomial for which the existence and uniqueness are proved and a one-to-one correspondence between equilibria and pairs of gas mass and liquid volume is established. We prove that these equilibria exhibit nonlinear and exponential asymptotic stability under small perturbations that conserve gas mass and liquid volume, and an equilibrium solution acts as a local minimizer of the energy functional, even under relatively large perturbations, with the proportionality constant determined by the adiabatic constant. Moreover, we construct a global center manifold to apply the center manifold theory. Our results apply to gases and liquids of all sizes. Furthermore, we derive the optimal exponential decay rate for small liquid volumes by analyzing the spectrum bounds of the associated linear operator and show that decreasing the gas mass or increasing the temperature can accelerate the convergence rate, a behavior not seen in unbounded liquid scenarios.@article{HLY25cvpde,
author ={Hao, Chengchun and Luo, Tao and Yang, Siqi},
journal={Calculus of Variations and Partial Differential Equations},
title ={Exponential stability of a free boundary problem with spherical symmetry for a gas bubble immersed in a bounded incompressible liquid},
year ={2025},
volume ={64},
number ={},
pages ={207 (45pp)},
doi={10.1007/s00526-025-03091-6},
} - Wei Zhang, Jie Fu, Chengchun Hao and Siqi Yang
Global well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximationThis paper focuses on the global well-posedness of the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial states of the drop are close to a ball $ B_{R}$ with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of $ B_{R}$. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck-Boussinesq approximation in a fixed domain. Then, we establish time-weighted estimates of solutions for the shifted equation using maximal $ L^p$-$ L^q$ regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang in 2022. Using the time decay estimates of the semigroup, we obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval $(0,T)$ was proven by Hao and Zhang in 2023. After that, we demonstrate that the solutions can be extended beyond $ T$ by analyzing the properties of the roots of algebraic equations.@article{ZFHY24a,
author ={Zhang, Wei and Fu, Jie and Hao, Chengchun and Yang, Siqi},
journal={Journal of Mathematical Physics},
title ={Global well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation},
year ={2024},
volume ={65},
number ={8},
pages ={081509 (22pp)},
doi={10.1063/5.0220764},
} - Chengchun Hao and Wei Zhang
Local well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximationThis paper is concerned with the local well-posedness of the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another fluid separated by a closed interface with surface tension. We devote to getting the linearized Oberbeck-Boussinesq approximation in the fixed domain by using the Hanzawa transformation, and using maximal $ L^{p}$-$ L^{q}$ regularities for the two-phase fluid motion of the linearized system obtained by the authors in [11] to establish the existence and uniqueness of the solutions of nonlinear problem with the help of the contraction mapping principle, in which the differences of nonlinear terms are estimated.@article{HZ23CPAA,
author ={Hao, Chengchun and Zhang, Wei},
journal={Communications on Pure and Applied Analysis},
title ={Local well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation},
year ={2023},
volume ={22},
number ={7},
pages ={2099-2131},
doi={10.3934/cpaa.2023059},
} - Chengchun Hao and Wei Zhang
Maximal $ L^p$-$ L^q$ regularity for two-phase fluid motion in the linearized Oberbeck-Boussinesq approximationThis paper is concerned with the generalized resolvent estimate and the maximal $ L^p$-$ L^q$ regularity of the linearized Oberbeck-Boussinesq approximation for unsteady motion of a drop in another fluid without surface tension, which is indispensable for establishing the well-posedness of the Oberbeck-Boussinesq approximation for the two incompressible liquids separated by a closed interface. We prove the existence of $\mathcal{R}$-bounded solution operators for the model problems and the maximal $ L^p$-$ L^q$ regularity for the system. The key step is to prove the maximal $ L^p$-$ L^q$ regularity theorem for the linearized heat equation with the help of the $\mathcal{R}$-bounded solution operators for the corresponding resolvent problem and the Weis operator-valued Fourier multiplier theorem.@article{HZ22JDE,
author ={Hao, Chengchun and Zhang, Wei},
journal={Journal of Differential Equations},
title ={Maximal $ L^p$-$ L^q$ regularity for two-phase fluid motion in the linearized Oberbeck-Boussinesq approximation},
year ={2022},
volume ={322},
number ={},
pages ={101-134},
doi={10.1016/j.jde.2022.03.022},
} - Jie Fu, Chengchun Hao, Siqi Yang and Wei Zhang
不可压新Hooke 型弹性动力学自由边界问题的爆破准则 NEW
(献给肖玲教授85华诞)
本文证明 3 维不可压新 Hooke 型弹性动力学模型自由边界问题解的 Beale-Kato-Majda 型爆破准则. 该结果表明, 在 Taylor 型符号条件成立的情形下, 只要形变张量和速度场的旋度、自由边界的第二基本形式、标准指数映射的单射半径以及压力的某些范数有界, 并且速度和形变张量的梯度及压力梯度的物质导数在自由边界上有界, 该解就可以一直延续.@article{fhyzb,
author ={Fu, Jie and Hao, Chengchun and Yang, Siqi and Zhang, Wei},
journal={中国科学：数学},
title ={不可压新Hooke 型弹性动力学自由边界问题的爆破准则},
year ={2025},
volume ={55},
number ={3},
pages ={635-650},
doi={10.1360/SSM-2023-0320},
} - Chengchun Hao and Dehua Wang
A priori estimates for the free boundary problem of incompressible neo-Hookean elastodynamicsA free boundary problem for the incompressible neo-Hookean elastodynamics is studied in two and three spatial dimensions. The a priori estimates in Sobolev norms of solutions with the physical vacuum condition are established through a geometrical point of view of Christodoulou and Lindblad (2000) [3]. Some estimates on the second fundamental form and velocity of the free surface are also obtained.@article{jde16_en,
author ={Hao, Chengchun and Wang, Dehua},
journal={Journal of Differential Equations},
title ={A priori estimates for the free boundary problem of incompressible neo-Hookean elastodynamics},
year ={2016},
volume ={261},
number ={1},
pages ={712–737},
doi={10.1016/j.jde.2016.03.025},
} - Chengchun Hao
Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densitiesIn this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $ \gamma=2$ and the general initial density in \(H^5\). Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $ L^3$.@article{Hao15,
author ={Hao, Chengchun},
journal={Discrete and Continuous Dynamical Systems - B},
title ={Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities},
year ={2015},
volume ={20},
number ={9},
pages ={2885-2931},
doi={10.1016/j.jde.2016.03.025},
} - Chengchun Hao and Hai-Liang Li
Well-posedness for a multidimensional viscous liquid-gas two-phase flow modelThe Cauchy problem of a multidimensional ($ d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is studied in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local-in-time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.@article{HL12SIMA,
author ={Hao, Chengchun and Li, Hai-Liang},
journal={SIAM Journal on Mathematical Analysis},
title ={Well-posedness for a multidimensional viscous liquid-gas two-phase flow model},
year ={2012},
volume ={44},
number ={3},
pages ={1304–1332},
doi={10.1137/110851602},
} - Chengchun Hao
Global well-posedness for a multidimensional chemotaxis model in critical Besov spacesWe investigate the Cauchy problem of a multidimensional chemotaxis model with initial data in critical Besov spaces. The global existence and uniqueness of the strong solution is shown for initial data close to a constant equilibrium state.@article{Hao12ZAMP,
author ={Hao, Chengchun},
journal={Zeitschrift für angewandte Mathematik und Physik},
title ={Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces},
year ={2012},
volume ={63},
number ={5},
pages ={825-834},
doi={10.1007/s00033-012-0193-0},
} - Yiquan Lin, Chengchun Hao and Hai-Liang Li
Global well-posedness of compressible bipolar Navier-Stokes-Poisson equationsWe consider the initial value problem for multi-dimensional bipolar compressible Navier-Stokes-Poisson equations, and show the global existence and uniqueness of the strong solution in hybrid Besov spaces with the initial data close to an equilibrium state.@article{LHL12AMSE,
author ={Lin, Yiquan and Hao, Chengchun and Li, Hai-Liang},
journal={Acta Mathematica Sinica, English Series},
title ={Global well-posedness of compressible bipolar Navier-Stokes-Poisson equations},
year ={2012},
volume ={28},
number ={5},
pages ={925-940},
doi={10.1007/s10114-011-0238-x},
} - Chengchun Hao
Well-posedness for the viscous rotating shallow water equations with friction termsWe consider the Cauchy problem for viscous rotating shallow water equations with friction terms. The global existence of the solution in some hybrid spaces is shown for the initial data close to a constant equilibrium state away from the vacuum.@article{Hao11JMP,
author ={Hao, Chengchun},
journal={J. Math. Phys.},
title ={Well-posedness for the viscous rotating shallow water equations with friction terms},
year ={2011},
volume ={52},
number ={2},
pages ={No.023101, 12pp},
doi={10.1063/1.3553187},
} - Chengchun Hao
Well-posedness to the compressible viscous magnetohydrodynamic systemThis paper is concerned with the Cauchy problem of the compressible viscous magnetohydrodynamic (MHD) system in whole spatial space $ \R^d$ for $ d\geqslant 3$. It is shown that the global solution exists uniquely in hybrid Besov spaces provided the initial data close to a constant equilibrium state away from the vacuum.@article{Hao11NARWA,
author ={Hao, Chengchun},
journal={Nonlinear Analysis: Real World Applications},
title ={Well-posedness to the compressible viscous magnetohydrodynamic system},
year ={2011},
volume ={12},
number ={6},
pages ={2962–2972},
doi={10.1016/j.nonrwa.2011.04.017},
} - Chengchun Hao
Cauchy problem for viscous shallow water equations with surface tensionWe are concerned with the Cauchy problem for a viscous shallow water system with a third-order surface-tension term. The global existence and uniqueness of the solution in the space of Besov type is shown for the initial data close to a constant equilibrium state away from the vacuum by using the Friedrich's regularization and compactness arguments.@article{,
author ={Hao, Chengchun},
journal={Discrete Contin. Dyn. Syst. Ser. B},
title ={Cauchy problem for viscous shallow water equations with surface tension},
year ={2010},
volume ={13},
number ={3},
pages ={593-608},
doi={10.3934/dcdsb.2010.13.593},
} - Chengchun Hao, Ling Hsiao and Hai-Liang Li
Cauchy problem for viscous rotating shallow water equationsWe consider the Cauchy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modeling of motions for shallow water with free surface in a rotating sub-domain Marche (2007) [19]. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without Coriolis forces, see for instance Danchin (2000) [10], Haspot (2009) [16], the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, and additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments.@article{HHL09JDE,
author ={Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang},
journal={Journal of Differential Equations},
title ={Cauchy problem for viscous rotating shallow water equations},
year ={2009},
volume ={247},
number ={12},
pages ={3234–3257},
doi={10.1016/j.jde.2009.09.008},
} - Chengchun Hao and Hai-Liang Li
Global existence for compressible Navier–Stokes–Poisson equations in three and higher dimensionsThe compressible Navier–Stokes–Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions.@article{HL09JDE,
author ={Hao, Chengchun and Li, Hai-Liang},
journal={Journal of Differential Equations},
title ={Global existence for compressible Navier–Stokes–Poisson equations in three and higher dimensions},
year ={2009},
volume ={246},
number ={12},
pages ={4791–4812},
doi={10.1016/j.jde.2008.11.019},
} - Recent progress on some kinds of nonlinear dispersive partial differential equations (in Chinese) Research Series for Excellent Doctoral Dissertation of Chinese Academy of Sciences vol. 2007, 1-6, Science Press, Beijing, 2008.
- Chengchun Hao, Ling Hsiao and Hai-Liang Li
Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational termIn this paper, we establish the global well posedness of the Cauchy problem for the Gross–Pitaevskii equation with a rotational angular momentum term in the space $\Bbb{R}^2$.@article{HHL08MMAS,
author ={Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang},
journal={Math. Methods Appl. Sci.},
title ={Global well posedness for the Gross–Pitaevskii equation with an angular momentum rotational term},
year ={2008},
volume ={31},
number ={6},
pages ={655-664},
doi={10.1002/mma.931},
} - Chengchun Hao, Ling Hsiao and Hai-Liang Li
Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensionsIn this paper, we establish the global well posedness of the Cauchy problem for the Gross-Pitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space $\Bbb{R}^3$.@article{HHL07JMP,
author ={Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang},
journal={J. Math. Phys.},
title ={Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions},
year ={2007},
volume ={48},
number ={10},
pages ={No.102105, 11pp},
doi={10.1063/1.2795218},
} - Chengchun Hao
Well-posedness for one-dimensional derivative nonlinear Schrödinger equationsIn this paper, we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form $ iu_t-u_{x x}+i\lambda |u|^k u_x=0$ with non-zero $\lambda\in \mathbb R$ and any real number $ k\geq 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $ H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.@article{Hao07CPAA,
author ={Hao, Chengchun},
journal={Commun. Pure Appl. Anal.},
title ={Well-posedness for one-dimensional derivative nonlinear Schrödinger equations},
year ={2007},
volume ={6},
number ={4},
pages ={997-1021},
doi={10.3934/cpaa.2007.6.997},
} - Chengchun Hao, Ling Hsiao and Baoxiang Wang
Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces@article{HHW07JMAA,
author ={Hao, Chengchun and Hsiao, Ling and Wang, Baoxiang},
journal={Journal of Mathematical Analysis and Applications},
title ={Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces},
year ={2007},
volume ={328},
number ={1},
pages ={58-83},
doi={10.1016/j.jmaa.2006.05.031},
} - Chengchun Hao, Ling Hsiao and Baoxiang Wang
Wellposedness for the fourth order nonlinear Schrödinger equationsWe study the local smoothing effects and wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in 1D $$ i\partial_t u=\partial_x^4 u+P\left((\partial_x^\alpha u)_{|\alpha|\leqslant 2}, (\partial_x^\alpha \bar{u})_{|\alpha|\leqslant 2}\right),\quad t,x\in \Bbb{R}, $$ where $ P(\cdot)$ is a polynomial excluding constant and linear terms.@article{HHW06JMAA,
author ={Hao, Chengchun and Hsiao, Ling and Wang, Baoxiang},
journal={Journal of Mathematical Analysis and Applications},
title ={Wellposedness for the fourth order nonlinear Schrödinger equations},
year ={2006},
volume ={320},
number ={1},
pages ={246-265},
doi={10.1016/j.jmaa.2005.06.091},
} - Baoxiang Wang, Chengchun Hao and Henryk Hudzik
Energy scattering theory for the nonlinear Schrödinger equations with exponential growth in lower spatial dimensionsFor one and two spatial dimensions, we show the existence of the scattering operators for the nonlinear Schrödinger equation with exponential nonlinearity in the whole energy spaces.@article{WHH06JDE,
author ={Wang, Baoxiang and Hao, Chengchun and Hudzik, Henryk},
journal={Journal of Differential Equations},
title ={Energy scattering theory for the nonlinear Schrödinger equations with exponential growth in lower spatial dimensions},
year ={2006},
volume ={228},
number ={1},
pages ={311-338},
doi={10.1016/j.jde.2006.05.010},
} - Chengchun Hao
The initial boundary value problem for quasi-linear Schrödinger-Poisson equationsIn this article, the author studies the initial-(Dirichlet) boundary problem for a high-field version of the Schrödinger-Poisson equations, which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schrödinger equations on the unit cube. A global existence and uniqueness is established for a solution to this problem.@article{Hao06AMSBE,
author ={Hao, Chengchun},
journal={Acta Math. Sci. Ser. B Engl. Ed.},
title ={The initial boundary value problem for quasi-linear Schrödinger-Poisson equations},
year ={2006},
volume ={26},
number ={1},
pages ={115-124},
doi={10.1016/S0252-9602(06)60033-7},
} - Chengchun Hao and Ling Hsiao
Studies on Schrödinger-Poisson systems (Excerpt of Dissertation)J. Grad. Sch. of CAS, 22(5): 141-146, 2005. bibThe bipolar(defocusing nonlinear)Schrödinger-Poisson system and quasi—linear Schrödinger—Poisson equations ale studied．The wellposedness，large time behavior and modified scattering theory is established for the Cauchy problem to the bipolar(defocusing nonlinear)Schrödinger—Poisson systems．The initial-(Dirichlet) boundary problem for a high field version of the Schrödinger-Poisson equations，quasi-linear Schrödinger-Poisson equations，which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schrödinger equations on the unit cube ale also discussed．@article{HH05JCAS,
author ={Hao, Chengchun and Hsiao, Ling},
journal={J. Grad. Sch. of CAS},
title ={Studies on Schrödinger-Poisson systems (Excerpt of Dissertation)},
year ={2005},
volume ={22},
number ={5},
pages ={141-146},
} - Chengchun Hao and Hai-Liang Li
On the initial value problem for the bipolar Schrödinger-Poisson systemsIn this paper，we prove the existence and uniqueness of global solutions in $ H^s(\Bbb{R}^3)$ ($ s\in \Bbb{R}，s\geqslant 0$)for the initial value problem of the bipolar Schrödinger-Poisson systems.@article{HL04JPDE,
author ={Hao, Chengchun and Li, Hai-Liang},
journal={J. Partial Diff. Eqns.},
title ={On the initial value problem for the bipolar Schrödinger-Poisson systems},
year ={2004},
volume ={17},
number ={3},
pages ={283-288},
} - Chengchun Hao, Ling Hsiao and Hai-Liang Li
Modified scattering for bipolar nonlinear Schrödinger-Poisson equationsIn this paper, we study the asymptotic behavior in time and the existence of the modified scattering operator of the globally defined smooth solutions to the Cauchy problem for the bipolar nonlinear Schrödinger–Poisson equations with small data in the space $ \R^3$.@article{HHL04M3AS,
author ={Hao, Chengchun and Hsiao, Ling and Li, Hai-Liang},
journal={Math. Models Methods Appl. Sci.},
title ={Modified scattering for bipolar nonlinear Schrödinger-Poisson equations},
year ={2004},
volume ={14},
number ={10},
pages ={1481-1494},
doi={10.1142/S0218202504003684},
} - Chengchun Hao and Ling Hsiao
Large time behavior and global existence of solution to the bipolar defocusing nonlinear Schrödinger-Poisson systemIn this paper, we study the large time behavior and the existence of globally defined smooth solutions to the Cauchy problem for the bipolar defocusing nonlinear Schrödinger-Poisson system in the space $\R^3$.@article{HH04QAM,
author ={Hao, Chengchun and Hsiao, Ling},
journal={Quart. Appl. Math.},
title ={Large time behavior and global existence of solution to the bipolar defocusing nonlinear Schrödinger-Poisson system},
year ={2004},
volume ={62},
number ={4},
pages ={701-710},
doi={10.1090/qam/2104270},
} - Chengchun Hao
Energy scattering for the generalized Davey-Stewartson equations@article{Hao03AMAS,
author ={Hao, Chengchun},
journal={Acta Mathematicae Applicatae Sinica, English Series},
title ={Energy scattering for the generalized Davey-Stewartson equations},
year ={2003},
volume ={19},
number ={2},
pages ={333-340},
doi={10.1007/s10255-003-0108-0},
} - Chengchun Hao
Energy Scattering for Nonlinear Davey-Stewartson EquationsJournal of Mathematical Research and Exposition, 23(4): 645-650, 2003. bib@article{Hao03JMRE,
author ={Hao, Chengchun},
journal={Journal of Mathematical Research and Exposition},
title ={Energy Scattering for Nonlinear Davey-Stewartson Equations},
year ={2003},
volume ={23},
number ={4},
pages ={645-650},
} - Chengchun Hao
Energy scattering for the generalized Davey-Stewartson equations (科技创新快报)Journal of Hebei University (Natural Science Edition), 22(1): 73-74, 2002. bib@article{Hao02JHBU,
author ={Hao, Chengchun},
journal={Journal of Hebei University (Natural Science Edition)},
title ={Energy scattering for the generalized Davey-Stewartson equations (科技创新快报)},
year ={2002},
volume ={22},
number ={1},
pages ={73-74},
} - Energy scattering for nonlinear Davey-Stewartson equations (in Chinese)
J. Math. Res. Expo. 23(4), 645-650, 2003.
In this paper, the author studies the scattering theory of a class of nonlinear Davey-Stewartson equations with three nonlinearities. The author proves that their scattering operator exists in the energy space $H^1$. - Energy scattering for the generalized Davey-Stewartson equations (Rapid Communications) Hebei Uni., Rat. Sci. Edt. 22(1), 73-74, 2002.
- Hai-Liang Li, Guo-Jing Zhang, Min Zhang and Chengchun Hao
Long-time self-similar asymptotic of the macroscopic quantum modelsThe unipolar and bipolar macroscopic quantum models derived recently, for instance, in the area of charge transport are considered in spatial one-dimensional whole space in the present paper. These models consist of nonlinear fourth-order parabolic equation for unipolar case or coupled nonlinear fourth-order parabolic system for bipolar case. We show for the first time the self-similarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a self-similar wave (which is not the exact solution of the models) in large time at an algebraic time-decay rate.@article{LZZH08JMP,
author ={Li, Hai-Liang and Zhang, Guo-Jing and Zhang, Min and Hao, Chengchun},
journal={J. Math. Phys.},
title ={Long-time self-similar asymptotic of the macroscopic quantum models},
year ={2008},
volume ={49},
number ={7},
pages ={No.073503, 14pp},
doi={10.1063/1.2949082},
} - Chengchun Hao, Yueling Jia and Hai-Liang Li
Quantum Euler-Poisson system: local existence of solutionsThe one-dim ensional tran sient quantum Euler-Poisson system for sem i-conductors iS studied in a bounded interva1．The quan tum correction can be interpreted as a dispersive regularization of the classical hydrodynamic equations and mechanical efects． The existence an d uniqueness of local-in-time solutions are proved with lower regularity an d without the restriction on the smallness of velocity，where the pressure—density is general(can be non—convex or non-monotone)．@article{HJL03JPDE,
author ={Hao, Chengchun and Jia, Yueling and Li, Hai-Liang},
journal={J. Partial Diff. Eqns.},
title ={Quantum Euler-Poisson system: local existence of solutions},
year ={2003},
volume ={16},
number ={4},
pages ={306-320},
}
Books or Lectures
Published or Accepted Papers
■ Free boundary problems in fluid dynamics equations
♣ MHD equations
♣ Two-phase equations
♣ Elastodynamics
♣ Euler equations
■ Well-posedness of Cauchy problems to fluid dynamics equations
■ Well-posedness of IVP/IBVP to dispersive equations
■ Well-posedness or asymptotic of IVP/IBVP to quantum models
2025-07-31
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